Putting It Together: Exponential and Logarithmic Functions

At the start of this module, you were considering investing your inheritance possibly to save for retirement. Now you can use what you’ve learned to figure it out. The final value of your investment can be represented by the equation

[latex]f(t)=Pe^{\large{rt}}[/latex]

where

[latex]P[/latex] = the initial investment

[latex]t[/latex] = number of years invested

[latex]r[/latex] = interest rate, expressed as a decimal

Now remember that you had $10,000 to invest, so [latex]P=10,000[/latex]. Also recall that the interest rate was 3%, so [latex]r=0.03[/latex].

Let’s start with 5 years, so [latex]t=5[/latex].

Start with the function: [latex]f(t)=Pe^{\large{rt}}[/latex]
Substitute P, r, and t: [latex]f(5)=10,000e^{\large{0.03}{(5)}}[/latex]
Evaluate: [latex]f(5)=11,618.34[/latex]

Now let’s look at 10 years, so [latex]t= 10[/latex].

Start with the function: [latex]f(t)=Pe^{\large{rt}}[/latex]
Substitute P, r, and t: [latex]f(10)=10,000e^{\large{0.03}{(10)}}[/latex]
Evaluate: [latex]f(10)=13,498.59[/latex]

Now let’s look at 50 years, so [latex]t=50[/latex].

Start with the function: [latex]f(t)=Pe^{\large{tr}}[/latex]
Substitute P, r, and t: [latex]f(50)=10,000e^{\large{0.03}{(50)}}[/latex]
Evaluate: [latex]f(10)=44,816.89[/latex]

Using the function for continuously compounded interest, you can see how your initial investment will grow over time.

[latex]t[/latex] Interest rate [latex]f(t)[/latex]
5 0.03 $11,618.34
10 0.03 $13.498.59
50 0.03 $44,816.89

Now you know that your $10,000 can grow to over $44,000 in 50 years! With that knowledge under your belt, you can decide if you want to add to your investment or find an account with a greater interest rate. Either way, thanks to your knowledge of exponential functions, you can make sound financial decisions.